Find Imaginary and real part

Question

Find out the imaginary and real parts, simple multiplications, but confused in the roots. Can any one give proper solution for it.

$$(\sqrt{3} + 2\sqrt{i})(3\sqrt{3}+2\sqrt{i})$$

This is how much I solved !

The number $3+2i$ has two square roots. Which one do you have in mind? — José Carlos Santos, Oct 31 '17 at 15:59
@NaDimKhAn: try to use the MathJax to avoid any confusion in your questions. — Amin, Oct 31 '17 at 16:03
The square root in on individuals @JoséCarlosSantos not on both, someone edited the question — Na Dim Kh An, Oct 31 '17 at 16:03
@NaDimKhAn: Check this page. https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference — Amin, Oct 31 '17 at 16:07
@NaDimKhAn It's clear now. Why don't you do multiplication? It's so easy. — José Carlos Santos, Oct 31 '17 at 16:10
I can multiply , i have solved some questions, but i am confused when the multiplication goes down to the roots — Na Dim Kh An, Oct 31 '17 at 16:11
@NaDimKhAn Please say what did you get and I (or someone else) will tell you if that's right or wrong. — José Carlos Santos, Oct 31 '17 at 16:17
ok, just simply answer me what will i get when i multiply, (2i)(3√3) & (2i)(2√i) — Na Dim Kh An, Oct 31 '17 at 16:20
You know, it's not so nice to demand from people to answer you, whatever your question is. José has been kind enough to offer some hints and has asked you politely to show your own work, so that he may point you in the right direction. — B. Pasternak, Oct 31 '17 at 16:21
I was actually working on a partial answer for this, but after reading through the comments, I have deleted the draft in opposition to @NaDimKhAn’s demand — gen-ℤ ready to perish, Oct 31 '17 at 16:27
"ok, just simply answer me what will i get when i multiply, (2i)(3√3) & (2i)(2√i)". You can $(6\sqrt{3})i$ for the first one and $4i\sqrt{i}$ for the second. However $\sqrt{i}$ is ambiguous and has two possible answers. Do you know how to find the two possible $(a + bi)$ where $(a+bi)^2 = i$? — fleablood, Oct 31 '17 at 17:20

score 0 · Accepted Answer · answered Oct 31 '17 at 17:16

$(\sqrt{3} + 2\sqrt{i})(3\sqrt{3}+2\sqrt{i})=$

$\sqrt{3}*3\sqrt{3} + 2\sqrt{i}*3\sqrt{3} + \sqrt{3}*2\sqrt{i} + 2\sqrt{i}*2\sqrt{i} = $.

$9\sqrt{3} + 6*\sqrt{3}*\sqrt{i} + 2*\sqrt{3}\sqrt{i} + 4*\sqrt{i}^2$ =

$= 9\sqrt{3} + 8\sqrt{3}\sqrt{i} + 4i$

Now we need to know what $\sqrt{i}$ is. Have you learned that?

Now I must make an editorial comment. $\sqrt{i}$ is very badly written. All non-zero complex or real numbers have two square roots and it is not at all clear which one is meant.

Let $(a+bi)^2 = a^2 + 2abi + b^2i^2 = (a^2 -b^2) + 2abi = i$.

So $a^2 - b^2 = 0$ and $2ab = 1$ and $a,b \in \mathbb R$. Can you solve this?

$a^2 = b^2$ and $2ab = 1$.

$a + bi = \sqrt{i}$ and there are two solutions. And in this case it is unclear which one they want.

So solve for $\sqrt{i}$ and plug it in.

You will have two possible answers.

Find Imaginary and real part

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